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Construction of Solitary Wave Solutions and Rational Solutions for mKdV Equation with Initial Value Problem by Homotopy Perturbation Method

DOI: 10.4236/oalib.1104383, PP. 1-10

Subject Areas: Partial Differential Equation

Keywords: mKdV Equation, Homotopy Perturbation Method, Soliton Solution

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Abstract

The mKdV equation with the initial value problem is studied numerically by means of the homotopy perturbation method. The analytical approximate solutions of the mKdV equation are obtained. Choosing the form of the initial value, the single solitary wave, two solitary waves and rational solutions are presented, some of which are shown by the plots.

Cite this paper

Dong, Z. and Wang, F. (2018). Construction of Solitary Wave Solutions and Rational Solutions for mKdV Equation with Initial Value Problem by Homotopy Perturbation Method. Open Access Library Journal, 5, e4383. doi: http://dx.doi.org/10.4236/oalib.1104383.

References

[1]  Chow, S.N., Mallet-Paret, J. and Yorke, J.A. (1978) Finding Zeros of Maps: Homotopy Methods That Are Constructive with Probability One. Mathematics of Computation, 32, 887-899.
https://doi.org/10.1090/S0025-5718-1978-0492046-9
[2]  Watson, L.T. (1986) Numerical Linear Algebra Aspects of Globally Convergent Homotopy Methods. SIAM Review, 28, 529-545. https://doi.org/10.1137/1028157
[3]  Li, T.Y. and Sauser, T. (1987) Homotopy Methods for Eigenvalue Problems. Linear Algebra and Its Applications, 91, 65-74.
https://doi.org/10.1016/0024-3795(87)90060-7
[4]  Liao, S.J. (1995) An Approximate Solution Technique Not Depending on Small Parameters: A Special Example. International Journal of Non-Linear Mechanics, 30, 371-380.
https://doi.org/10.1016/0020-7462(94)00054-E
[5]  He, J.H. (2004) Comparison of Homotopy Perturbation Method and Homotopy Analysis Method. Applied Mathematics and Computation, 156, 527-539.
https://doi.org/10.1016/j.amc.2003.08.008
[6]  Ganji, D.D. and Sadighi, A. (2006) Application of He’s Homotopy-Perturbation Method to Nonlinear Coupled Systems of Reaction-Diffusion Equations. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 411-418.
https://doi.org/10.1515/IJNSNS.2006.7.4.411
[7]  Javidi, M. and Golbabai, A. (2007) A Numerical Solution for Solving System of Fredholm Integral Equations by Using Homotopy Perturbation Method. Applied Mathematics and Computation, 189, 1921-1928.
https://doi.org/10.1016/j.amc.2006.12.070
[8]  Wang, Q. (2007) Homotopy Perturbation Method for Fractional KdV Equation. Applied Mathematics and Computation, 190, 1795-1802.
https://doi.org/10.1016/j.amc.2007.02.065
[9]  Odibat, Z. (2007) A New Modification of the Homotopy Perturbation Method for Linear and Nonlinear Operators. Applied Mathematics and Computation, 189, 746-753.
https://doi.org/10.1016/j.amc.2006.11.188
[10]  Abbasbandy, S. (2007) Application of He’s Homotopy Perturbation Method to Functional Integral Equations. Chaos Solitons & Fractals, 31, 1243-1247.
https://doi.org/10.1016/j.chaos.2005.10.069
[11]  Chen, Z.Y., Huang, N.N., Liu, Z.Z. and Xiao, Y. (1993) An Explicit Expression of the Dark N-Soliton Solution of the MKdV Equation by Means of the Darboux Transformation. Journal of Physics A: Mathematical and General, 26, 1365-1374.
https://doi.org/10.1088/0305-4470/26/6/018
[12]  Zeng, Y.B. and Dai, H.H. (2001) Constructing the N-Soliton Solution for the mKdV Equation through Constrained Flows. Journal of Physics A: Mathematical and General, 34, L657-L663.
https://doi.org/10.1088/0305-4470/34/46/103
[13]  Inc, M. (2007) An Approximate Solitary Wave Solution with Compact Support for the Modified KdV Equation. Journal of Applied Mathematics and Computing, 184, 631-637.
https://doi.org/10.1016/j.amc.2006.06.062
[14]  Wazwaz, A.M. (2007) The Extended Tanh Method for Abundant Solitary Wave Solutions of Nonlinear Wave Equations. Journal of Applied Mathematics and Computing, 187, 1131-1142.
https://doi.org/10.1016/j.amc.2006.09.013
[15]  Wazwaz, A.M. (2008) New Sets of Solitary Wave Solutions to the KdV, mKdV, and the Generalized KdV Equations. Communications in Nonlinear Science and Numerical Simulation, 13, 331-339.
https://doi.org/10.1016/j.cnsns.2006.03.013
[16]  Gao, Y.F., Dai, Z.D. and Li, D. (2009) New Exact Periodic Solitary-Wave Solution of mKdV Equation. Communications in Nonlinear Science and Numerical Simulation, 14, 3821-3824.
https://doi.org/10.1016/j.cnsns.2008.09.011
[17]  Yasar, E. (2010) On the Conservation Laws and Invariant Solutions of the mKdV Equation. Journal of Mathematical Analysis and Applications, 363, 174-181.
https://doi.org/10.1016/j.jmaa.2009.08.030
[18]  Parkes, E.J. (2010) Observations on the Tanh-Coth Expansion Method for Finding Solutions to Nonlinear Evolution Equations. Journal of Applied Mathematics and Computing, 217, 1749-1754.
https://doi.org/10.1016/j.amc.2009.11.037
[19]  Zhang, J.B., Zhang, D.J. and Shen, Q. (2011) Bilinear Approach for a Symmetry Constraint of the Modified KdV Equation. Journal of Applied Mathematics and Computing, 218, 4494-4500.
https://doi.org/10.1016/j.amc.2011.10.030
[20]  Trogdon, T., Olver, S. and Deconinck, B. (2012) Numerical Inverse Scattering for the Kortewegde Vries and Modified Kortewegde Vries Equations. Physica D, 241, 1003-1025.
https://doi.org/10.1016/j.physd.2012.02.016
[21]  Yin, J.L., Xing, Q.Q. and Tian, L.X. (2014) Complex Dynamical Behaviors of Compact Solitary Waves in the Perturbed mKdV Equation. Chinese Physics B, 23, 080201.
https://doi.org/10.1088/1674-1056/23/8/080201
[22]  Xin, X.P., Miao, Q. and Chen, Y. (2014) Nonlocal Symmetry, Optimal Systems, and Explicit Solutions of the mKdV Equation. Chinese Physics B, 23, 010203.
https://doi.org/10.1088/1674-1056/23/1/010203

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